\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

↓

\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_* \cdot x\right))_*}{2} \cdot \sin y i\right))\]

double f(double x, double y) {
        double r2246252 = x;
        double r2246253 = exp(r2246252);
        double r2246254 = -r2246252;
        double r2246255 = exp(r2246254);
        double r2246256 = r2246253 + r2246255;
        double r2246257 = 2.0;
        double r2246258 = r2246256 / r2246257;
        double r2246259 = y;
        double r2246260 = cos(r2246259);
        double r2246261 = r2246258 * r2246260;
        double r2246262 = r2246253 - r2246255;
        double r2246263 = r2246262 / r2246257;
        double r2246264 = sin(r2246259);
        double r2246265 = r2246263 * r2246264;
        double r2246266 = /* ERROR: no complex support in C */;
        double r2246267 = /* ERROR: no complex support in C */;
        return r2246267;
}

↓

double f(double x, double y) {
        double r2246268 = x;
        double r2246269 = exp(r2246268);
        double r2246270 = -r2246268;
        double r2246271 = exp(r2246270);
        double r2246272 = r2246269 + r2246271;
        double r2246273 = 2.0;
        double r2246274 = r2246272 / r2246273;
        double r2246275 = y;
        double r2246276 = cos(r2246275);
        double r2246277 = r2246274 * r2246276;
        double r2246278 = 0.016666666666666666;
        double r2246279 = 5.0;
        double r2246280 = pow(r2246268, r2246279);
        double r2246281 = 0.3333333333333333;
        double r2246282 = r2246268 * r2246268;
        double r2246283 = fma(r2246281, r2246282, r2246273);
        double r2246284 = r2246283 * r2246268;
        double r2246285 = fma(r2246278, r2246280, r2246284);
        double r2246286 = r2246285 / r2246273;
        double r2246287 = sin(r2246275);
        double r2246288 = r2246286 * r2246287;
        double r2246289 = /* ERROR: no complex support in C */;
        double r2246290 = /* ERROR: no complex support in C */;
        return r2246290;
}

\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))

↓

\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_* \cdot x\right))_*}{2} \cdot \sin y i\right))

Derivation

Initial program 43.7
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
Taylor expanded around 0 0.8
\[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2} \cdot \sin y i\right))\]
Simplified0.9
\[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left(x \cdot (\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right))_*}}{2} \cdot \sin y i\right))\]
Final simplification0.9
\[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_* \cdot x\right))_*}{2} \cdot \sin y i\right))\]

Error

Derivation

Reproduce